1 Input-Output Mappings. 2 Hebbian Failure. 3 Delta Rule Success.
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1 Task Learnng 1 / 27 1 Input-Output Mappngs. 2 Hebban Falure. 3 Delta Rule Success.
2 Input-Output Mappngs 2 / Output Input Make approprate: Response gven stmulus. Interpretaton of a stuaton. Expectaton of what happens next. Plan for sequence of future actons.
3 Three Input/Output Mappngs 3 / 27 Easy Hard Event_2 Event_3 Event_2 Event_3 Event_0 Event_1 Imposs Event_0 Event_1 Event_2 Event_3 Event_0 Event_1
4 Task Learnng: Mnmzng Error (Gradent Descent) Task error = Summed-Squared Error: SSE = (t k o k ) 2 (1) t To mnmze the error, take the dervatve of the error wth respect to the weghts: ndcates how error changes as weghts change. Delta Rule mnmzes SSE: k w k = ɛ(t k o k )s (2) 4 / 27
5 Credt/Blame Assgnment 5 / 27 a) b) Weghts reflect strongest soluton (vs. strongest correlaton n Hebban).
6 Example: Mnmzng y = x 2 va dervatves 6 / y = x negatve steep dervatve (x= 2) (x=1) postve shallow dervatve x How does y change w/changes to x? Dervatve of y wrt x; dy y dx or x. Dervatve of x 2 = 2x. To mnmze y, move x opposte the dervatve.
7 Mnmzng Error 2 SSE 7 / 27 SSE = t k (t k o k ) 2, dervatve: w k = ɛ(t k o k )s a) t k=.5 o =s w = w k k k wk b) w ) (.5 k (negatve) = dsse dwk (postve) s =1 0 (zero) w k.5
8 Dervaton of Delta for Lnear Unts SSE = t k (t k o k ) 2. Actvaton: o k = s w k SSE w k = SSE o k o k w k (3) SSE o k = 2(t k o k ) (4) o k w k = s (5) SSE w k = 2(t k o k )s (6) w k = ɛ(t k o k )s (7) 8 / 27
9 Delta Rule for Bas Weghts 9 / 27 Bas Wts: Treat sendng unt as always actve at 1: β k = ɛ(t k o k ) (8)
10 Summary and Further Issues 10 / 27 We can mnmze SSE for unts wth lnear acts usng delta rule! 1 What about sgmodal/pont neuron actvatons? Use cross-entropy error (CE) = delta rule w/sgmodal acts. Later we ll get to pont neuron actvatons. 2 What s the target value, really? Target = outcome phase of actvaton. 3 Delta rule weghts are unbounded need to bound 0-1.
11 Cross-Entropy Error (vs. SSE) 11 / 27 Cross-Entropy assumes t k, o k are probabltes of bnary vars: CE = t k log o k + (1 t k ) log(1 o k ) (9) t Bg penalty f o k = 0 and t k = 1: k Sum Squared vs Cross Entropy Error CE SSE Error Output Actvaton (Target = 1)
12 CE Cancels out Dervatve wth Sgmodal Unts 12 / 27 CE o k CE w k = CE o k do k dη k η k w k (10) = t k o k (1 t k) (1 o k ) = t k o k o k (1 o k ) (11) do k dη k = σ (η k ) = o k (1 o k ) (12) η k w k = s (13) CE w k = (t k o k )s (14)
13 What s Target? Actvaton Phases 13 / 27 a) Mnus Phase (expectaton) output b) Plus Phase (outcome) target Input Input w k = ɛ(o + k o k )s (15)
14 Soft Weght Boundng 14 / 27 Keep weghts bounded between 0-1 by exponentally slowng ncreases, decreases as they approach bounds: w k = [ k ] + (1 w k ) + [ k ] w k (16) [ k ] + = computed weght change f postve (else 0). [ k ] = computed weght change f negatve (else 0).
15 Task Learnng II: Revenge of the Hdden layer 15 / 27 1 Impossble tasks and hdden layers. 2 Generalzed Delta Rule: Backpropagaton. 3 Bologcally plausble verson: GeneRec.
16 Re-representng and Hdden layers 16 / 27 Dffcult tasks become easer when you re-represent usng ntermedate representatons: Memorze dgts usng dgt chunks. Read n terms of words, not letters. Hdden layers enable ths re-representaton! (multple levels of transformatons). Delta rule can t do ths.
17 Error Backpropagaton Actvaton 17 / 27 Propagate error sgnals to hdden unts so they can adust weghts: a) Output Hdden Input b) Output Error Hdden Input Targets o k = σ(η ) k η k= Σh w k h η =Σ t k s o k = σ(η ) s w w k = (t k o k ) h h w =? s
18 Bp: The Equatons Actvaton 18 / 27 Error Output Hdden Input Targets t k o k h s δ k = ( t k o k ) w k = δ k h δ =Σ δ k w k w = δ s σ ( η ) w = ɛδ x (17) For output unts: δ k = (t k o k ) For hdden unts: δ = ( k δ kw k ) ( h (1 h ) )
19 Bp: The Dervaton 19 / 27 Maor chan rule: CE w = k dce do k do k dη k η k h dh dη η w (18) Compare to delta rule: CE w k = CE o k do k dη k η k w k (19) It took years (and sufferng through Mnsky & Papert) to add those 2 extra chan steps!
20 The Problem wth BP Actvaton 20 / 27 How does that δ get propagated backwards across the synapse, down the axon, and out the dendrtes?? Error Output Hdden Input Targets t k o k h s δ k = ( t k o k ) w k = δ k h δ =Σ δ k w k w = δ s σ ( η )
21 GeneRec: Bologcally Plausble Bp 21 / 27 Use bdrectonally-connected network wth 2 phases of settlng: a) Mnus Phase b) Plus Phase (actual output) External Target o k t k w k w k w k w k h h + w w s s External Input External Input
22 GeneRec: The Equatons 22 / 27 o k w k = (t h k o k ) h w = ( h+ h ) s s External Input Learnng rule s same as delta rule! w = ɛ(y + y )x (20)
23 GeneRec: Where the Error Comes From 23 / 27 Hdden unts get actual output (o k ) and target (t k ) sgnals va actvaton propagaton (net nput) from output layer: η = s w + k o k w k (21) η + = s w + k t k w k (22) Subtract the two net nputs to get δ : ( η + η = s w + ) ( t k w k s w + k k = t k w k o k w k k k o k w k ) = k (t k o k )w k δ (23)
24 GeneRec: Implct Actvaton Dervatve Actvaton 24 / 27 In Bp, δ = k (t k o k )w k h (1 h ), and h (1 h ) s dh dη. If we compute y + y nstead of η + η, then we get dh dη free! GeneRec works wth any actvaton functon, ncl. pont neuron! ( η + η ) σ (η) σ (η) h h + σ(η) η η+ Net nput
25 Symmetry + Mdpont = CHL 25 / 27 Need weghts to be symmetrc, and why should we use x for the sendng unt actvaton nstead of x +?? Take the average of the sendng and recevng weght updates, and use the average of the plus and mnus phases for the sendng unt: w = ɛ 1 [ (x x )(y + y ) + (y + [ ] = ɛ (x + ) (x y y + ) ] + y )(x + x ) Ths s the Contrastve Hebban Learnng rule (CHL) what we actually use! (24)
26 Bologcal Implementaton LTD LTP 26 / 27 Plus Phase x +, y + 0 x +, y + 1 Mnus Phase Err CPCA Combo Err CPCA Combo x, y x, y Just lke CPCA except when you make an error: x, y 1 then x +, y + 0. LTD here comes from moderate amount of Ca 2+ n mnus phase. Θ Θ + 2+ [Ca ]
27 Nature of the Tranng Sgnals 27 / 27 a) Explct Teacher b) Implct Expectaton Hdden Hdden Input Output Output Input Outcome Outcome t t+.5 t+1 t t+.5 t+1 c) Implct Motor Expectaton d) Implct Reconstructon Hdden Hdden Motor Outcome Outcome Input Input Input t t+.5 t+1 t t+.5 t+1
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